This title slide is titled "Dijkstra's Algorithm: Finding the Shortest Path." It includes the subtitle "A Beginner's Guide for Operations Research Students."

Dijkstra's Algorithm: Finding the Shortest Path

A Beginner's Guide for Operations Research Students

Source: Operations Research Presentation

This slide introduces weighted graphs, using cities as nodes and roads as edges connecting them. Weights on the edges represent distances between cities.

Introduction to Weighted Graphs

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• Nodes represent cities or locations.
• Edges are roads connecting the cities.
• Weights indicate distances on those roads.

Source: Dijkstra's algorithm

The "Why Shortest Path Matters" slide highlights how shortest path algorithms save time and fuel in navigation, like Google Maps. It also covers optimizing delivery routes for Amazon logistics and speeding up internet data routing.

Why Shortest Path Matters

• Saves time and fuel (e.g., Google Maps navigation)
• Optimizes delivery routes (e.g., Amazon logistics)
• Speeds up network routing (e.g., internet data)

The slide explains Dijkstra’s algorithm via a maze analogy on the left: starting at S, greedily selecting the closest unvisited spot, and updating paths until all nodes are reached. The right side previews a small weighted graph with green-highlighted S, arrows to nodes like A=3 and B=5, and an animation showing distance updates.

Understanding Dijkstra’s Algorithm

Source: Operations Research Presentation

This agenda slide, titled "Algorithm Overview," outlines four steps of a shortest-path algorithm. It covers initializing distances (source=0, others=∞), selecting the unvisited node with smallest distance, updating neighbors' distances, and repeating until all nodes are visited.

Algorithm Overview

1. 1. Initialize distances

Start with S=0, set all others to infinity (∞). Graph icon shows initial state.

2. 2. Pick smallest dist unvisited node

Select the unvisited node with the smallest current distance. Graph icon highlights it.

3. 3. Update neighbors' distances

Check and update distances to its unvisited neighbors if shorter. Graph icon updates.

4. 4. Mark visited, repeat till done

Mark node as visited, repeat steps 2-3 until all nodes visited. Graph icon marks. Source: Dijkstra's Algorithm for Beginners

The slide titled "Step 1: Initialization" displays a table with columns for Node, Dist (distance), and Prev (predecessor). Node S is initialized with distance 0 and no predecessor, while nodes A, B, C, and D all have unknown distances and no predecessors.

Step 1: Initialization

{ "headers": [ "Node", "Dist", "Prev" ], "rows": [ [ "S", "0", "-" ], [ "A", "unknown", "-" ], [ "B", "unknown", "-" ], [ "C", "unknown", "-" ], [ "D", "unknown", "-" ] ] }

Source: Dijkstra's Algorithm Example

In Step 2 Iteration 1, node S is visited (green), updating distances to A=10 via S and B=5 via S, with the table showing S|0|- (visited), A|10|S, B|5|S, C|∞|-, D|∞|-. Next, B is picked for its smallest distance of 5, and its neighbors will be relaxed.

Step 2: Iteration 1

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• S visited (green), updates: A=10 via S, B=5 via S.
• Table: S|0|- (visited), A|10|S, B|5|S, C|∞|-, D|∞|- .
• Pick B (smallest distance=5), relax its neighbors.

Source: Dijkstra's Algorithm Visualization

In Iteration 2, node B is selected with distance 5 from S and marked visited. Distances are updated to C (8 from B) and D (13 from B), with the table showing S|0|, B|5|S(visited), A|10|S, C|8|B, D|13|B.

Step 3: Iteration 2

!Image

• Select B (distance 5), mark visited.
• Update C: min(∞, 5+3)=8 from B.
• Update D: min(∞, 5+8)=13 from B.
• Table: S|0|, B|5|S(visited), A|10|S, C|8|B, D|13|B.

Source: Dijkstra's algorithm

In Iteration 3, node A is selected with the smallest distance of 10 from S. Node C's distance remains 8 via B (no change), D updates to 12 via A, yielding table A|10|S, C|8|B, D|12|A and path S-A-D.

Step 4: Iteration 3

!Image

• Select A (smallest distance = 10)
• C: min(8, 10+1) = 8 (no change)
• D: min(13, 10+2) = 12 (updated via A)
• Table: A|10|S, C|8|B, D|12|A; path S-A-D

Source: Dijkstra's algorithm

The slide "Step 5: Final Shortest Paths Found" displays a table of shortest distances from source node S to other nodes. It lists: S (distance 0, path S), B (5, S), A (10, S), C (8, B), and D (12, A).

Step 5: Final Shortest Paths Found

{ "headers": [ "Node", "Shortest Distance", "Path" ], "rows": [ [ "S", "0", "S" ], [ "B", "5", "S" ], [ "A", "10", "S" ], [ "C", "8", "B" ], [ "D", "12", "A" ] ] }

Source: Dijkstra's Algorithm Presentation

The "Real-World Applications" slide features a grid of four real-world uses for optimal routing algorithms. It covers Google Maps navigation for shortest paths, logistics truck deliveries for efficiency, internet data packet routing for speed, and AI character paths in video games.

Real-World Applications

{ "features": [ { "icon": "🗺️", "heading": "Google Maps Routes", "description": "Finds shortest paths for navigation, saving time on daily drives." }, { "icon": "📦", "heading": "Logistics Deliveries", "description": "Optimizes truck routes to deliver packages faster and cheaper." }, { "icon": "🌐", "heading": "Internet Data Packets", "description": "Routes data efficiently across networks for speedy web access." }, { "icon": "🎮", "heading": "Games AI Paths", "description": "Guides characters via optimal routes in virtual game worlds." } ] }

This conclusion slide summarizes weighted graphs as networks with distances, Dijkstra's algorithm for finding shortest paths from a start, and its applications in maps, logistics, and networks, with the final graph revealing full paths. It encourages trying it yourself, noting that the basics are mastered—explore more!

Summary

• Weighted graph: network with distances

• Dijkstra's finds shortest paths from start
• Crucial for maps, logistics, networks
• Final graph reveals full paths

Try it yourself! 😊

Basics mastered—explore more!

Source: Dijkstra's Algorithm Presentation